"There is no universal agreement about whether to include zero in the set of natural numbers: some define the natural numbers to be the positive integers {1, 2, 3, ...}, while for others the term designates the non-negative integers {0, 1, 2, 3, ...}. The former definition is the traditional one, with the latter definition having first appeared in the 19th century."
https://en.wikipedia.org/wiki/Natural_number

Not sure if I understand the concept of this question. But I remember the following words from my Professor "First thing, you always do when you buy a boot about Mathematics, check on their definition of \(\mathbb{N}\)"
What he meant by that, was that whether or not \(0 \in \mathbb{N}\) or not is still a big discussion. For references check for example Zorich Analysis 1, Blatter Analysis and so on. Also consider the notation \(\mathbb{N}_0\)

The footnote of this quotation was as follows:
"Given two elements N and E, say, we can construct a field by the rules N+N=N, N+E=E, E+E=N, N\(\cdot\)N=N, N\(\cdot\)E=N, E\(\cdot\)E=E. Then, in keeping with our notation, we should write N=0, E=1 and hence 2=1+1=0. To exclude such number systems, we require that all natural field elements be nonzero"
Basically, I don't know what's going on here.

Warning: Please ignore the following.
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Should 0 be included in the set of natural numbers?
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<Saying I> No, \(0\notin \mathbb N\)
The numbers 1, 1+1=2, 2+1=3, ... are said to be natural.
Given two elements N and E, we can struct a field by the rules
(i) N+N=N
(ii) N+E=E
(iii) E+E=N
(iv) N\(\cdot\)N = N
(v) N\(\cdot\)E = N
(vi) E\(\cdot\)E = E
Then in keeping with our notation, we should be N=0, E=1.
From (iii), E+E = N, so, we have 1+1 = 0
However from the first statement, we have 1+1=2 \(\ne\) 0, so we need to exclude 0.
Problems:
1) Why would we construct such a field with rules (i) to (vi), particularly with rule (iii)?
2) What would happen if we picked some other values for N and E?